B I F U R C A T I O N S I N T W O - D I M E N S I O N A L R E V E R S I B L E M A P S T . P O S T a n d H . W . C A P E L l n s t i t u u t v o o r T h e o r e t i s c h e F y s i c a , U n i v e r s i t e i t v a n A m s t e r d a m , V a l c k e n i e r

Physica A 164 (1990) 625-662 North-Holland

B I F U R C A T I O N S IN T W O - D I M E N S I O N A L R E V E R S I B L E M A P S

T . P O S T a n d H . W . C A P E L

lnstituut voor Theoretische Fysica, Universiteit van Amsterdam, Valckenierstraat 65, 1018 X E Amsterdam, The Netherlands

G . R . W . Q U I S P E L l

Department o f Theoretical Physics, Research School o f Physical Sciences, The Australian National University, Canberra, A C T 2601, Australia and Mathematics Research Section, School o f Mathematical Sciences, The Australian National University, Canberra, A C T 2601, Australia

J . P . V A N D E R W E E L E

Centrum voor Theoretische Natuurkunde, Universiteit Twente, Postbus 217, 7500 A E Enschede, The Netherlands

Received 15 November 1989

We give a treatment of the non-resonant bifurcations involving asymmetric fixed points with Jacobian J ~ 1 in reversible mappings of the plane. These bifurcations include the saddle- node bifurcation not in the neighbourhood of a fixed point with J = 1, as well as the so-called transcritical bifurcations and generalized Rimmer bifurcations taking place at a fixed point with Jacobian J = 1. The bifurcations are illustrated by some simple examples of model maps.

The Rimmer type of bifurcation, with e.g. a center point with J = 1 changing into a saddle with Jacobian ~,~ = 1, an attractor and a repeller, occurs under more general conditions, i.e.

also in non-reversible mappings if only a certain order of local reversibility is satisfied. These Rimmer bifurcations are important in connection with the emergence of dissipative features in non-measure-preserving reversible dynamical systems.

I. Introduction

R e v e r s i b l e m a p s c a n b e a n a d e q u a t e t o o l f o r t h e d e s c r i p t i o n o f p h y s i c a l s y s t e m s t h a t d i s p l a y c o n s e r v a t i v e b e h a v i o u r i n c e r t a i n p a r t s o f t h e i r p h a s e s p a c e , a n d d i s s i p a t i v e b e h a v i o u r i n o t h e r p a r t s . T h e s e t w o k i n d s o f b e h a v i o u r , c o e x i s t i n g i n t h e s a m e p h y s i c a l s y s t e m , h a v e b e e n d i s c u s s e d i n r e f s . [ 1 - 7 ] .

Present address Department of Mathematics, La Trobe University, Bundoora, Victoria 3083, Australia.

0378-4371/90/$03.50 © Elsevier Science Publishers B.V.

(North-Holland)

626 1 ~. Post et al. / B~[i~rcations in 2 D reversible maps

A m a p T is called reversible if there exists a d e c o m p o s i t i o n

T= I I L . (1.1)

w h e r e I~ and 12 a r e involutions, i.e. m a p s which are their o w n inverse,

1~ 1-~ = 1 . (1.2)

T h e involutions Ii, /2 a r e often called symmetries of the m a p T, because for each invariant set M o f the m a p T: T o M = M, we can easily show that the

"'reflections" I~ o M a n d 1~ o M arc also invariant u n d e r T.

( F r o m T M = IFI~M = M we have 12M = I , M and h e n c e TI~M = lll21tM = It1212M = l I M and TI2M = 111212M = IEM = LM. [E)

As a typical e x a m p l e we study the t w o - p a r a m e t e r m a p (cf. ref. [8])

[

a : ' = x + c o ( y ) ( m o d l ) , K

T" Y' ) , + ~ g(.~, ) (1..%)

,

with g ( x + 1 ) = g ( x ) and h ( x + l ) = h ( x ) . With the choices c o ( y ) = y , g(x) sin(2-rrx), we r e c o v e r for E = 0 the C h i r i k o v - T a y l o r s t a n d a r d m a p [9]. If c o ( y ) = -co(y) the m a p p i n g is reversible with the involutions

x ' x ( m o d l ) ,

I," - y + ~ g(x)

K

1 + eyh(x)

(1.4)

x ' = x + c0(y) ,

1~" y, = - Y . (1.5)

T h e s y m m e t r y curve associated with 1~ is

[eh(x)]y'-

+ 2 y - ~

K

g(x) = 0 , (1.6)

while the s y m m e t r y curve associated with 12 is simply

y = 0 .

(1.7)

T h e s y m m e t r i c fixed points of the m a p p i n g are given by y , = 0 and g(xo) O.

If g(x) and h(x) are b o t h o d d the m a p is reversible with the d e c o m p o s i t i o n

T. Post et al. / Bifurcations in 2 D reversible maps

627

T =

1112

given by

I

x' = - x (mod 1 ) ,

11:

y - ~

K

g(x)

(1.8)

x = - x - o J ( y ) ,

12: y ' y . (1.9)

In this case we have the symmetry curves 2x E Z of I 1 and 2x + w(y) ~ Z of 12.

T h e symmetric fixed points are thus given by 2x 0 E Z and o~(y0)E Z.

The fixed points of the map (1.3) with

g(Xo)=

0 and Y0 = 0 trivially have Jacobian J = 1. The other fixed points can be divided into two types labeled FP1 and FP2, as follows:

FP1:

Kg(xo)=eh(xo)=O

and

W ( y o ) @ Z ;

FP2:

K - 2 _ h(xo)

Kg(xo)~O

and

eh(xo)~O

and

2~e Yo g(Xo)

and

w(yo) E Z .

H e r e the FP2 are always asymmetric fixed points with j ~ 1. The FP1 have Jacobian J = 1. They are symmetric fixed points in the case that both

g(x)

and

h(x)

are odd, they are asymmetric fixed points when oJ(y) is odd, and

g(x)

and

h(x)

are not both odd.

In section 2 we give a stability analysis of the FP1 and FP2. In section 3 we give a description of the non-resonant bifurcations of the map (1.3), i.e. the bifurcations at which FP2 type fixed points can be born, either by a s a d d l e - node bifurcation [10], not in the neighbourhood of one of the FP1, or by a transcritical bifurcation [10] or a R i m m e r bifurcation [11-13], taking place at one of the FPI's. Each bifurcation type is illustrated using appropriate choices for

w(y), g(x)

and

h(x).

Finally, in section 4, we discuss the concept of local reversibility, introduced in ref. [14], and the relations with the various bifurca- tion scenarios.

2. Stability analysis

For the elements of the Jacobian matrix of the map (1.3) we find

628 T. Post et al. / Bifurcations in 2D reversible map.~

Ox' OX'

- 1 , - ~o'( y ) ,

a x a y

t(

a y ' 2~r g ' ( x ' ) + e y y ' h ' ( x ' ) a y ' a y ' a x 1 - e y h ( x ' ) a y - j: + c o ' ( y ) a x

(2.1)

w h e r e the Jacobian ~¢ is given by

l + e v ' h ( x ' ) v' ' - ~ g(x')

. _ . _ _ .

(" ~)

¢ - l - E y h ( x ' ) v K " "

"

+ g ( x ' ) /

For a f i x e d p o i n t ( x . , Yo) we have

_ _

K

t . 2 t

(2.3)

K

1 + ey,~h(x,,) Y o - Fvv g(x,,)

51"' I e y , , h ( x , , ) - K (2.4)

Y" + 2rr g(x,,)

T r , , = l + J + c o ' ( ( O r '

(2.5)

w h e r e Try, is the trace of the J a c o b i a n matrix in (x o, y , ) . T h e stability ^of the fixed points is g o v e r n e d by the eigenvalue e q u a t i o n

a 2 + a T r . + ¢ . 0 , (2.6)

in particular, we have a saddle if [Tro] > 1 + j'~,.

F o r the FP1 we have ,c/~ 1 and h e n c e (x,,. y~,) is a s a d d l e if

, (y,,) o y ' ) > 0 ( or o , ' ( y o ) , ( O y ' } J

< 4 ( 2 . 7 )

3X /o o

and (x o, Yo) is a c e n t e r if

- 4 < ~o'( y,,) ( aY' ), < o

.

(2.s)

For the FP2 we have ,¢o ~ 1 and (xc~ , y . ) is a s a d d l e if

T. Post et al. / Bifurcations in 2D reversible maps

629

( O y ' ]

> 0 or

w ' ( y o ) ( O Y ' ~

< - 2 ( 1 - J o ) (2.9)

~°'(Y°) \

Ox /o \ Ox /o "

Otherwise (x0, Yo) is an

attractor

(~0 < 1) or a

repeller

(¢0 > 1).

For future use we observe that

(Oy'/OX)o

in the case of a FP2 can be written as (cf. eq. (2.3))

( Oy' ] _ ey~g'(Xo) ( h'(xo) h(xo)

Ox ]o 1 - - - ~ )

\g'(xo) g(Xo))"

^(2.10)

3. Bifurcation types

We shall now analyze the bifurcation scenarios of the FP2. F r o m the definition of the FP2 type fixed points we observe that for reasonable w ( y ) the fixed point condition w(y0) E Z can only be satisfied for a discrete set of values Yo. This leads us to consider the condition

h(xo) K -2

g(Xo) - 2~r6 Yo

(3.1)

for fixed values of Y0.

For our purposes it is now convenient to consider a plot of z =

h(x)/g(x)

in which also the horizontal line z =

-(K/2"rre)yo 2

is drawn, The position of the latter line obviously depends on the map parameters K and E. A t the intersections with z =

h(x)/g(x)

the fixed point condition (3.1) is satisfied. The bifurcation scenarios which emerge are described below.

3.1. Saddle-node bifurcation

At a maximum of

h(x)/g(x)

at x = Xma x (where we have no FP1) we have a situation as depicted in fig. 1. Clearly three situations can be distinguished:

K -2 h(Xmax)

(a) 2"rre Yo > g(Xmax---- ~ "

There are no FP2 in the n e i g h b o u r h o o d of X ma x.

K - 2 _ h(xmax) (b) 2~rE Y0 g(Xmax) •

There is one FP2 at x = x 0 = Xma x.

6 3 0 T. Post et al. / Bifurcations in 2 D reversible maps

(a) -K -2

( l ) x = h(x)

z g(x)

Fig. 1. The function z h ( x ) / g ( x ) near x,,~. Also drawn are lines z = (K/2qxe)y. e, for the three situations (a), (b) and (c) discussed in the text. The two fixed points of type FP2 in situation (c) arc denoted by x,,(l) and x.(r).

K , 2 h ( x . . . ) (c) 2 w e 5'' < g ( x ... ) '

T h e r e a r e t w o F P 2 , o n e w i t h x 0 < x ... a n d o n e w i t h x~ > x ... . W e i n v e s t i g a t e t h e i r s t a b i l i t y b y c o n s i d e r i n g

3 . 2 )

I

W e s e e t h a t o o ' ( y o ) ( a y ' / O x ) ~ ~ c h a n g e s sign at x = x ... , so in s i t u a t i o n (c) we h a v e

- e i t h e r a n a t t r a c t o r ( A ) a n d a s a d d l e ( S ) w i t h f < I if e y o h ( x ... ) < ( 1 (cf.

e q . ( 2 . 4 ) ) ,

- o r a r e p e l l e r ( R ) a n d a s a d d l e ( S ) w i t h f > 1 if e y o h ( x ... ) > ( / .

I n s i t u a t i o n ( b ) t h e s i n g l e F P 2 h a s o n e e i g e n v a l u e e q u a l to 1, a n d t h e o t h e r s m a l l e r o r l a r g e r t h a n 1, a g a i n d e p e n d i n g o n t h e sign o f e y o h ( x ... ). H e r e t h e s a d d l e - n o d e b i f u r c a t i o n t a k e s p l a c e .

T h e c a s e t h a t h ( x ) / g ( x ) h a s a m i n i m u m i n s t e a d o f a m a x i m u m c a n b e t r e a t e d a n a l o g o u s l y .

A s a n e x a m p l e o f this b i f u r c a t i o n w e c o n s i d e r t h e m a p ( 1 . 3 ) w i t h t h e c h o i c e s

t o ( y ) = y , g ( x ) = c o s ( 2 c r x ) , h ( x ) = c o s ( 4 r r x ) , ( 3 . 3 )

t h a t is,

x ' = x + y ( m o d l ) , T : v + K c o s ( 2 - r r x ' )

} 2 t z "

" 1 - e y c o s ( 4 v x ' )

( 3 . 4 )

T. Post et al. / Bifurcations in 2 0 reversible maps

We then have

and

-

^{0.05 0 0.05}

X

-

Fig. 2. Saddle-node bifurcation. The figures are phase portraits of the example map (3.4). (a) At K = 0.32 and r = -0.05. Here K is slightly below the bifurcation value -27~6. This is the situation (a) in the text where there are no FP2 in the neighbourhood of ( 0 , l ) . (b) At K = 0.31 and r = -0.05. Here K is slightly above the bifurcation value -27re. This is the situation (c) in the text.

There are two FP2, an attractor (A) and a saddle (S) with

9

< 1.

so the maxima and minima of h(x)lg(x) are at x =

0

(mod 1) and at x =

a

(mod I ) , respectively.

Taking

yo

=

1

we have the bifurcation at x,,, = 0 at

K

= -2ne and the bifurcation at x,,, =

4

^at

K

⁼ 2ne. In fig. 2 the bifurcation at x = 0 is illus- trated.

1

^1.03

y 1 , 0 2 1 . 0 1 1 0 . 9 9 0.98 0 , 9 7

a

,5555

/

- -- - ^--z45+,'

/--

,---

j----

/ ,

_ - - - ^{- /C/}

_{,, /} ^/

i

_{\ \ \}

' _L'--T=--

_'

.

_{\ - -}

-_

_{'\ l}

^--..

^\ _\

-I----

632 T. Post et al. I Bff'urcations in 21) reversible maps

3.2. Transcritical bifurcation

I f t h e r e is a F P 1 a t x = x ~ , w h i c h is n o t a m a x i m u m o r a m i n i m u m o f

/ '

h ( x ) / g ( x ) w e h a v e f o r p o s i t i v e ( h ( x ) g(x)),=,, o n e o f t h e s i t u a t i o n s d e p i c t e d in fig. 3. A t x = x~ w e h a v e h ( x ~ ) = g(x~) = 0 a n d t h e f u n c t i o n h ( x ) / g ( x ) h a s t h e v a l u e

lim h ( x ) _ h ' ( x , ) (3.7)

. , ~ , g ( x ) g ' ( x l) "

T h e t h r e e s i t u a t i o n s t o b e d i s t i n g u i s h e d a r e K h ' ( x , )

(a) 27re y 2 > g'(Yl~--)

W e h a v e a F P 1 a t x = x 1 a n d a F P 2 a t x x . ( a ) > x~.

K ~ h ' ( x I ) ( b ) 27re y'' g ' ( x , )

W e o n l y h a v e a f i x e d p o i n t a t x = x~.

K ~ h'(x~) (c) 2~rE y''~ < g'(x,~

W e h a v e a F P 1 a t x = Xl a n d a F P 2 a t x = &~(c) < x ~ . T h e s t a b i l i t y o f t h e F P I Xl f o l l o w s f r o m

(oy,)

°°'(Y°) \ Ox ix, = w ' ( Y ° ) 2ww g ' ( x , ) + ey~h (x 1

,o'( 2 , ( K

= - y i O e y o g ( x , ) - 2 w e y~)

h'(x~_) )

y,,'(x, ) "

(3.8)

(a) (b) (c)

z

T

h(x) x z (g(x}=h(XjLI=8) _z = g(x'-"~

g

Fig. 3. The function z = h(x)/g(x) near a FPI (at x x~ ), at which there is neither a maximum nor a minimum. Also drawn are lines z = -(K/2we)y~i "-. The fixed points of type FP2 in situation (a) and (c) are denoted by x,,(I) and x,,(r), respectively.

T. Post et al. / Bifurcations in 2D reversible maps

633

This q u a n t i t y c h a n g e s sign at the bifurcation, so t h e FP1 at x = x I c h a n g e s f r o m a saddle into a c e n t e r or vice versa w h e n the F P 2 at x = x 0 passes t h r o u g h t h e FP1.

O n the o t h e r h a n d , the stability o f the F P 2 x 0 follows f r o m

to (Yo) eYog (Xo)

( Oy' ] = ' z , ( h,(xo) h(xo ) )

to'(Yo) \ Ox /,, i Z - - e ~ \ g ' ( X o ) g ( X o ) / "

^(3.9)

N e a r x = x I we m a k e the e x p a n s i o n s

K - 2 _ h(xo)

2rr6

Yo g(xo)

-

h'(xl) +

(h"(Xl)

g ' ( X l ) -

h'(xl) g"(xl)) (x o xl) +

h . o . t . ,

g'(x~---~ \ - 2 ( g ' ~ "

(3.10)

and

h'(xo) _ h ' ( x , ) + (h"(Xl) g ' ( x , ) - h ' ( x 1)

g"(Xl) ) (x ° - X l ) + h . o . t . ",

g'(x°~ g'(Xl~ ~ 1 ~ ] ~

(3.11)

so close to the bifurcation (situation (b)) eq. (3.9) can be written as

(Oy' I , 2 , ( K -2 h ' ( x l ) ]

= to ( y o ) e y o g (x~) - ~ Yo

to'(Yo) \ Ox /o g ' ( x l ) /

^(3.12)

This is the s a m e as eq. (3.8) b u t with the o p p o s i t e sign.

F u r t h e r m o r e the J a c o b i a n of the F P 2 at x = x 0, which can be e x p a n d e d as

Jo = 1 + 2 e y o h ' ( X l ) (xo - x l ) +

h . o . t . , (3.13)

c h a n g e s f r o m values J < 1 to J > 1 at the bifurcation.

R e s u m i n g the f o r e g o i n g c o n s i d e r a t i o n s we c o n c l u d e that in going f r o m situation (a) to situation (c):

- either a c e n t e r ( C ) and a saddle (S) with J > 1 c h a n g e into a saddle with J = 1 (S) and an a t t r a c t o r (A) or vice versa,

- or a c e n t e r ( C ) and a saddle (S) with J < 1 c h a n g e into a saddle with J = 1 (S) a n d a repeller ( R ) or vice versa.

A s an e x a m p l e of this bifurcation we c o n s i d e r the m a p (1.3) with the choices

w ( y ) = y , g(x)

= sin(2~rx) - 1 ,

h(x)

= sin[2a'r(x - ~ ) ] , (3.14)

634 T. Post et al. / BiJurcations" in 2D reversible map,s

t h a t is,

x ' = x + y ( m o d 1 ) , T : y + ~ K s i n ( 2 ' r r x ' ) -

Y' 1 - e y sin[2nv(x' - i~ )]

I

2 ( 3 . 1 5 )

= J ( m o d l ) , so W e t h e n h a v e x I ~2

l i m h(x) 2 ~ / ~ ( 3 . 1 6 )

, - ~ g ( x ) 3 a n d

l i m ( h ( x ) ] ' 2

~ x , \ g ( x ) / = 3 ~ ' " ( 3 . 1 7 )

T a k i n g Y0 = 1 we h a v e t h e b i f u r c a t i o n at

K - 4 q'rX/-3 ~ . ( 3 . 1 8 )

I n fig. 4 this b i f u r c a t i o n is i l l u s t r a t e d .

3.3. R i m m e r bifurcation

If t h e r e is a FP1 at x = x j , at w h i c h t h e r e is a m a x i m u m o r a m i n i m u m o f h ( x ) / g ( x ) we h a v e a s i t u a t i o n as d e p i c t e d in fig. 5 ( f o r a m a x i m u m ) . A t x = x~

t h e f u n c t i o n h ( x ) / g ( x ) h a s a g a i n t h e v a l u e

l i m h ( x ) _ h ' ( x ~ ) ( 3 . 1 9 )

. . . . , g(x) g'(x, )

a n d t h e t h r e e s i t u a t i o n s to b e d i s t i n g u i s h e d a r e t h e s a m e as t h o s e o f t h e p r e v i o u s s u b s e c t i o n , n a m e l y :

K 2 h'(x~)

(a) 2"rre y'' > g'(xL~ "

K ~ h ' ( x l ) ( b ) 2"rrE Yo- g ' ( x , ) '

K h'(x~)

(c) 2~E y t 2 < g ' ( x t ) .

T. Post et al. / Bifurcations in 2 D reversible maps 635

I 1,982

g 1,991

i

9,393

9,~98

9,97 9,975 9,98 9.985 9,99 9,995 9,1

x >

~,992 U

1,991

1

9,998

ib

.... i .... i .... i , , • .. ... .... ... ... .

9.97 9,975 9,98 9,985 9,9~ 9,995 9,1

x )

Fig. 4, Transcritical bifurcation. T h e figures are p h a s e p o r t r a i t s of the e x a m p l e m a p (3.15). (a) A t K = - 0 . 3 7 and • = 0.05. H e r e K is slightly b e l o w the bifurcation value - ~ w ~ , / 3 e. We have an elliptic fixed point at x = ~ and a saddle with J > 1. (b) A t K = - 0 . 3 5 5 and • = 0.05. H e r e K is slightly a b o v e the bifurcation value - ~ , / 3 • , We have a saddle with J = 1 at x = ~2 and an attractor.

Fig. 5. T h e function z = h(x)/g(x) n e a r a m a x i m u m at x = x~, which is a fixed point of type FP1.

A l s o d r a w n are lines z = - ( K / 2 " ~ e ) y g 2. T h e two fixed points of type FP2, xo(1) and x0(r ), are b o r n f r o m the FP1 (situation (b)) as we go f r o m (a) to (c).

636 T. Post el al. / Bifurcation,s in 2 D reversible maps

In situations (a) and (b) we only have a FP1 at x x,, while in situation (c) we have two FP2 at both sides of xj as well.

T h e stability of the FP1 xj is again described by eq. (3.8). which reads ' )

',"(y0) 7~-x ,,

K e h'(x~)

= - o o ' ( y , , ) e y ~ g ' ( x , ) - ~ Yo g ( ' ~ ) , (3.20)

As already mentioned, this quantity changes sign at the bifurcation, so the FP1 at x = x~ changes f r o m a saddle into a center or vice versa here.

On the other hand. the stability of the FP2 x. is described by eq. (3.9).

reading

(av') o,'(

_Y0) e ) o g (~o)

,-~

^-

(tL'(x,,) /,(x,,)

) (3.21)

O u r task is again to expand h ( x o ) / g ( x , ) and h ' ( x , , ) / g ' ( x , , ) in the neighbour- hood of x = x t, as in eqs. (3.10) and (3.11). Only now we must do it up to second order in ( x , , - x t ). since we are dealing with an e x t r e m u m in this case.

We obtain

K ~ h(x,,)

2~E y'' g(x,,)

h'(x,) g'(x,)

~[ (h'/'(Xi) ~.~'(Xl)- h'(.'(,) gtt'(Xl))(x, ' j(,)2 } h.o.[.

)l - =

(3.22) and

h'(x~,) _ h ' ( x , ) + ( h ' " ( x 1 ) g ' ( x l ) , - h ' ( x , ) g " ' ( x , ) ) ( x . x,)~ + h . o . t . "

g'(x,,) g'(.,) g(g 7 ^13.23)

so close to the bifurcation (situation (b)) eq. (3.21) can be written as

, = e y o g ( x l ) - y,, •

- , o x i , ,

(3.24)

This is the same as eq. (3.20) multiplied by - 2 .

F u r t h e r m o r e we have the expansion of the Jacobian of the FP2 of the previous subsection, reading

J o = 1 + 2 e y o h ' ( x , ) ( x , - x l ) + h.o.t. (3.25)

T. Post et al. / Bifurcations in 2D reversible maps 637

T h e J a c o b i a n is seen to c h a n g e sign at x = x~, so o n e o f the two F P 2 has J < 1 while the o t h e r has J > 1.

R e s u m i n g the f o r e g o i n g c o n s i d e r a t i o n s we c o n c l u d e that in going f r o m situation (a) to situation (c):

- either a c e n t e r ( C ) c h a n g e s into a saddle (S) with J = 1, giving birth to an a t t r a c t o r (A) a n d a repeller ( R ) ,

- o r a saddle with J = 1 (S) c h a n g e s into a c e n t e r ( C ) , giving birth to a saddle (S) with J < 1 and a saddle (S) with J > 1.

T h e s e two possibilities are schematically illustrated in fig. 6 below.

As an e x a m p l e of this bifurcation we consider the m a p (1.3) with the choices

w ( y ) = y , g(x) = sin(2~rx) , h ( x ) = sin(4~rx), (3.26)

that is,

x ' = x + y ( m o d 1 ) , T: y + ~ K sin(2~rx')

y ^{r z}

1 - e y sin(4"rrx')

(3.27)

We t h e n simply have

h(x)

_ 2 c o s ( 2 7 r x ) , ( 3 . 2 8 )

g(x)

so the m a x i m a a n d m i n i m a of h ( x ) / g ( x ) are at the FP1 x = 0 ( m o d 1) a n d x = I ( m o d 1), respectively.

T a k i n g Yo = 1 we h a v e the bifurcation o f Xma x = 0 at K = -4-rr6 and the bifurcation at Xmi n = ½ at K = 47re. In figs. 7 and 8 the t w o possible bifurcations at x = 0 are illustrated.

It s h o u l d be n o t e d that in the case of a reversible m a p with a s y m m e t r i c fixed point, e.g. the m a p (1.3) with g ( x ) and h(x) b o t h o d d , the t w o newly b o r n fixed points m o v e off the s y m m e t r y lines. This generalizes the bifurcation in

S (Jac'Cl)

C < S

^(.~:~,

C .~ $

^(.~--I,

f -

R $ (Jac>1)

Fig. 6. Schematical representation of the Rimmer bifurcation. A center is indicated by (C), a saddle by (S), an attractor by (A) and a repeller by (R).

638 T. Post et al. ,I Bifurcations in 2 l) reversible maps

I 1,8882

Y 1,81181

11,5555

11,5558 -11,111

\ \ \ , " _ / , '

- 11,11115 8 8,1185

X

11,111 )

1,911112 Y

i,1111111

l

11,5555

i i ~ _ - - _ _ _ r ~

f

" " . . . . . . . . . . / /

- 0 , 8 1 - 8,8115 8 0 , 1 1 8 5 8.111

x - - - ~

Fig. 7. R i m m e r b i f u r c a t i o n . T h e figures a r e p h a s e p o r t r a i t s of t h e e x a m p l e m a p (3.27). (a) A t K = - 1 . 6 2 8 6 a n d • = 0.05. H e r e K is s l i g h t l y b e l o w t h e b i f u r c a t i o n v a l u e 4"rr•. W e h a v e a c e n t e r at x = 0. (b) A t K - - 0 . 6 2 8 a n d • - 0.05. H e r e K is s l i g h t l y a b o v e t h e b i f u r c a t i o n v a l u e - 4 r e . W e h a v e a s a d d l e w i t h ¢ = 1 a n d an a t t r a c t o r a n d a r e p e l l e r . In b o t h figures (a) a n d (b) t h e fixed p o i n t s a r e s u r r o u n d e d by K A M c u r v e s .

reversible area-preserving maps studied by R i m m e r [11-13] to the case of reversible non-area-preserving maps. However, the bifurcation depicted in figs.

6 - 8 can also occur under more general conditions, if the FP1 is an asymmetric fixed point of a reversible mapping, or even if the mapping is not reversible.

For convenience we will call this a R i m m e r bifurcation as well. In the next section the conditions for these more general Rimmer bifurcations will be investigated and related to the concept of local reversibility introduced in ref.

[14].

The R i m m e r bifurcations have been observed earlier in conservative sys- tems, e.g. in the appearance of two period-6 orbits in the H6non map [11], and

T. Post et al. / Bifurcations in 2 D reversible maps 639

I 1,e992 u

1,9991

I" •!:•i::•• ••• '• •• ...

: j

:~ ~< .

9 9,995 9,91

x )

1

9,99~9

9,~998

, , , , , , i

- 9,91 - 9,995

b " . . . " i ' . . . ..: . . . .!

I 1,nee2 .

... . . . ..!

1 - i ' .i x ×

e,,js~js ...

9,.. i i ,

• i

- 9,91 - 9,995 9 9,995 ,91

X

Fig. 8. R i m m e r b i f u r c a t i o n . T h e figures a r e p h a s e p o r t r a i t s of t h e e x a m p l e m a p (3.27). (a) A t K = 0.6286 a n d E = - 0 . 0 5 . H e r e K is s l i g h t l y a b o v e t h e b i f u r c a t i o n v a l u e -4-rre. W e h a v e a s a d d l e w i t h 5 t = 1 at x = 0. (b) A t K = 0.628 a n d E = - 0 . 0 5 . H e r e K is s l i g h t l y b e l o w t h e b i f u r c a t i o n v a l u e -4~xE. W e h a v e a c e n t e r at x = 0, a n d t w o s a d d l e s , o n e w i t h ~ < 1 a n d t h e o t h e r w i t h ~ > 1. T h e c e n t e r in (b) is s u r r o u n d e d b y K A M c u r v e s .

in period-2 orbits in a P o i n c a r 6 section associated with a set of reversible n o n - H a m i l t o n i a n differential e q u a t i o n s arising f r o m laser physics [4, 5]. In the s e c o n d case dissipative f e a t u r e s including the a p p e a r a n c e o f an a t t r a c t o r a n d a repeller have b e e n o b s e r v e d .

3.4. Dissipative features

In c o n n e c t i o n with the a b o v e scenarios it s h o u l d be m e n t i o n e d t h a t fixed points of type F P 2 can only o c c u r u n d e r certain conditions. F o r instance w h e n we consider

g(x)

a n d

h(x)

such that

h(x)/g(x)

has a well-defined m a x i m u m in [0, 1], it is clear that F P 2 ' s at a definite Y0 ~ 0, ~O(yo)E Z, can only o c c u r for

640 T. Post et al. / Bifltrcations in 2 D reversible maps

large enough values of E / K . This means that when we start from the conserva- tive mapping (1.3) with e = 0, K ~ 0, and perturb the mapping into a non- conservative mapping with • ¢ 0, K ~ 0, the FP2's will not occur for sufficiently small E.

In the special case that both g ( x ) and I t ( x ) arc odd, the mapping is reversible and m o r e o v e r the F P I ' s are symmetric fixed points. This implies in particular that K A M curves may be observed in the n e i g h b o u r h o o d of these fixed points, such as e.g. depicted in fig. 7a. Therefore, the non-conservative mapping (1.3) with • ~ 0 wil behave like a conservative system around the fixed point for sufficiently small •. Only at a relatively large value of

I•1,

such as e.g. the values • = + K / 4 T r , asymmetric fixed points with ,j ~ 1 will be born, c.f. fig. 7b, in which the dissipative features are clearly visible.

4. Local reversibility

In the present section we briefly expose the concept of local reversibility [141 and we discuss the implications concerning the various bifurcation scenarios described above. One can ask oneself under which conditions these can occur when one is studying the fixed points of some mapping of the plane.

A useful tool is then a local reversibility analysis, that is, one figures out to what extent the map can be written l o c a l l y near a fixed point as the product of two involutions W and V, which leave the fixed point invariant,

T = W V , (4.1)

where the involution W is taken to be

W = S K U S , (4.2)

S being an arbitrary coordinate transformation and U being the involution

I

^{X ~ X ,}

U: y, - Y . (4.3)

This way to write W implies hardly any loss of generality since locally such as S exists, namely S = U + IV, provided that U + W is locally invertible.

For this purpose one expands the mapping near a fixed point (x~, y~) in powers of w = x - x~ and z = y - Yl, up to a certain order,

~ w ' = A w + B z + E w ~- + F w z + a z ' - + L w : ' + M w ~ z + N w z ' - + O z ~ + . . . , T : ~ z = C w + D z + H w ~ + J w z + K z 2 + P w ~ + Q w 2 z + R w z ~- + S z 3 + . . . .

(4.4)

T. P o s t et al. / B i f u r c a t i o n s in 2 D reversible m a p s 641

Next one finds out what the necessary conditions on the coefficients A , B , . . . are under which one is able to find an involution V and a coordinate transformation S, such that the relation

S T = U S V (4.5)

is satisfied up to nth order in w and z. If the mapping meets these conditions it is called locally reversible u p to o r d e r n around (x l, Yl)-

For instance, for fixed points that lie on a symmetry line in a reversible map we obviously have local reversibility up to order ~.

The concept of local reversibility can be used as a negative criterion to show that a given mapping is n o t reversible. Consider e.g. the situation that (x~, Yl) is a fixed point such that it cannot be transformed into any other of the fixed points by an involution. This is in particular the case when the eigenvalues of the linearized mapping at any fixed point are different from the inverse eigenvalues of (x~, Yl). Then (x~, y~) should be a symmetric fixed point if the mapping is to be reversible. T h e r e f o r e , if in that case local reversibility is not satisfied at any finite order, it can be concluded that the mapping is not reversible, see ref. [14] for an example related to the map (1.3).

In table I, which is taken from ref. [14], we list the conditions for local reversibility up to third order. The first order condition, which is not in the table, is simply

J ( X l , Y l ) = 1. (4.6)

Table I

Necessary conditions for local reversibility of a m a p p i n g L given by the expansion (4.4), after a linear transformation to one of the Jordan n o r m a l forms. T h e first c o l u m n gives the Jordan n o r m a l form of the linear part of L in the various cases. In the second column the necessary conditions at second order are specified. T h e third order conditions are given in the third column. This table is t a k e n from ref. [14].

Normal form O r d e r 2 O r d e r 3

(A A0) no con tions

'1 B ) l either

,0 J + 2 E - 2 B H = 0

o r

H = 0

no conditions

/io I -"1)

(A - 1 ) ( A R + A - 1 M - F J - 2 G H ) + E F ( A 2 _ 2 A 1 ) + j K ( Z A 2 _ A 3 ) = 0

if H = 0 t h e n

3 B 2 p - B Q - 3 B L + 2 E K + E F + 6 B E 2 = 0 6 B 2 P - 2 B Q - 6 B L + 2 E K

+ 6 B E 2 + B E J + 2 B J 2 - F J - J K = 0

1 2 B P + 4 Q + 12L + 2 J 2 + 2 E J + 2 2 B E H + 10BZH 2 + I O F H + I 1 B H J + 4 H K + 1 2 E 2 = 0

642 7". P o s t et al. / B i f u r c a t i o n s in 2 D reversible m a p s

The three rows of the table thus correspond to the three possible Jordan normal forms that the linear part of the expanded mapping can be brought into. For a more extended discussion of table I we refer to ref. [14].

It may be concluded that the s a d d l e - n o d e bifurcations of subsection (3.1) correspond to zero-order local reversibility at the fixed points. A p a r t from that we only have to consider the local reversibility conditions at the F P I ' s . In the following we will show that in order to have the Rimmer type of bifurcation at a FP1 type fixed point the local reversibility should be satisfied up to second order. On the other hand, in order to have the transcritical type of bifurcation only first order local reversibility is required. This requirement is automatically satisfied for a FP1 though.

4.1. L o c a l a n a l y s i s at the F P 1

In order to work out the conditions for local reversibility, we expand the map (1.3) around a FP1 up to third order in the local coordinates w = x - .r~, z = y - y ~ , i.e.

~o(y) = b z + oJ~z 2 + w3z ~ , (4.7)

g ( x ) = g l w + g 2 w 2 + g3w ~ , (4.8)

h ( x ) = h t w + h , w 2 + h ~ w -~ . (4.9)

In this way we obtain a local map of which the linear part is given by

w' 1 b ' w \

with K

A = ~ gl + E y ~ h l . (4.11)

We refer to appendix A for some details of the derivation of this form and also of the quadratic and cubic parts, i.e. the higher order coefficients E . . . S.

The eigenvalue equation which follows from (4.10) is

(A - 1) 2 = A b A , (4.12)

where A denotes the eigenvalue. The FP1 is elliptic for

ITrl < 2,

that is

- 4 < Ab < 0 , (4.13)

and it is hyperbolic if Ab < - 4 or Ab > 0.

T. Post et al. / Bifurcations in 2 D reversible maps 643

At Ab = - 4 we have the single eigenvalue A = - 1 , meaning that the elliptic tixed point period-doubles here. At Ab = 0 we have the single eigenvalue ,4 = + 1. This is precisely the situation for which a bifurcation of the transcriti- cal type or the R i m m e r type as t r e a t e d in subsections (3.2) and (3.3) could occur. T h e linear matrix in this case reduces to

i.e. the second J o r d a n normal f o r m of table I. ( T h e period-doubling case with Ab = - 4 corresponds to the third J o r d a n normal form of table I.)

4.2. Second order conditions

To evaluate the conditions for local reversibility in the case A = 0 one simply reads the coefficients from the m a p p i n g (4.4) (given in appendix A, in eqs.

(A.7), (A.8)) and works out the conditions given in table I. O n e then obtains for second order local reversibility the condition that either

J 4- 2 E - 2 B H = 2Eyoh'(xl) = O, (4.15)

o r

2 , (h(x))'

H = eyog ( x , ) \ g ( x ) / x = ~ = 0. (4.16)

T h e first condition, which is also the condition for second order local measure preservation [14] is in general not satisfied. It m a y only give rise to a bifurcation at a FP1 in the case that h ' ( x l ) = g ' ( x l ) = 0 , which we will not investigate in detail.

The second condition states though that the FP1 (x 1, y~) must be at an e x t r e m u m of h(x)/g(x), which is exactly the requirement for a R i m m e r bifurcation to occur.

Thus the conditions A = 0, H = 0 are the conditions for a R i m m e r bifurcation in the m a p (1.3). These conditions hold m o r e generally for any m a p p i n g of the form (4.4), with general coefficients, with a linear part given by

that is, for h = 0, H = 0 one can have a bifurcation as schematically depicted in fig. 6. Some details of the derivation of the conditions for the m a p p i n g with general coefficients are given in appendix B.

644 T. Post el al. / Bifurcations in 2 D reversible maps

In the case that A ~ 0. that is, e i t h e r b e f o r e or a f t e r the R i m m e r bifurcation t a k e s place, t h e r e is no c o n d i t i o n on the coefficients for s e c o n d o r d e r local reversibility.

4.3. T h i r d o r d e r c o n d i t i o n s

In the case that A - 0 the third o r d e r c o n d i t i o n s can again bc e v a l u a t e d directly f r o m eqs. ( A . 7 ) , ( A . 8 ) . F o r H - 0 we then lind the simple condition

h"(x~ ) - g " ( x , ) = O , (4.17)

of. a p p e n d i x A, s u b s e c t i o n A . 2 . F o r H ~ 0 t h e r e is no condition.

F o r the case A ~ 0 the e x p a n s i o n (4.4) a r o u n d the F P I in the case of the m a p p i n g (1.3) m u s t be t r a n s f o r m e d such that its linear p a r t t a k e s the lirst n o r m a l f o r m of table I.

( A ( ' ) (1 A i .

T h e result of this (see a p p e n d i x A . s u b s e c t i o n A . 3 ) is that the condition for third o r d e r local reversibility of table I can be w o r k e d ()tit to give

, h ( x )

]'

,, 0. i41 t

T h u s it follows that the third o r d e r condition for A # 0 in the case of a FP1 of the m a p p i n g (1.3) is a u t o m a t i c a l l y implied by the s e c o n d o r d e r condition for a = 0. This, h o w e v e r , is not g e n e r a l l y the case for a m a p p i n g of the type (4.4) with general coefficients.

4.4. E x a r n p l e map

T o check the a b o v e results we study the m a p (1.3) with the choices w ( y ) = y + % ) , : ,

g ( x ) - sin(2~rx) + 8 sin2(2"rrx).

h ( x ) = sin(4"rrx) + ~ sin-~(4-rrx ') , that is,

l

,~-' x + v+to2y-" (rood 1 ) ,

T: y + ~ K s i n ( 2 r r x ' ) + 8 s i n e ( 2 v x ') 1 - Ey sin(4"rrx') + ½~/sin2(4,rrx'~

(4.19)

(4.2o)

T. Post et al. / Bifurcations in 2 D reversible maps

which is locally reversible up to second order near the FP1

(x~,y~)= 0, 1 + but

n o t

to third order if

~ , ~ 0 ,

~:=-q+ A,

645

(4.21)

(4.22) (4.23)

0,~9~2

0,~0~1

0,5858

-8,Or -fl,685 8 n,SO5 0,81

X

9,9932 b

9,9659

-8,81 -B,885 O 8,885 8,91

x

Fig. 9. R i m m e r b i f u r c a t i o n in a n o n - r e v e r s i b l e m a p . T h e f i g u r e s a r e p h a s e p o r t r a i t s of t h e e x a m p l e m a p (4.20) w i t h w 2 = 0 . 1 1 , 7 = 0 . 1 a n d 6 = - q + A . ( R e c a l l t h a t A = K + 4 ~ E y ~ . ) (a) A t K = - 0 . 5 1 9 5 a n d E = 0.05. H e r e A ~ - 2 . 2 8 5 x 10 ~. W e h a v e a c e n t e r at x = 0. (b) A t K = - 0 . 5 1 9 a n d E = 0.05. H e r e A ~ + 2 . 7 1 5 x 10 ~. W e h a v e a s a d d l e w i t h ~ = 1 a n d a n a t t r a c t o r a n d a r e p e l l e r . (a) r e p r e s e n t s o n e s i n g l e o r b i t , w h i l e (b) r e p r e s e n t s t w o orbits.

646 T. Post et al. BiJi~rcations in 2 D reversihle tnap~

cf. eq. (4.17) for the case a = 0 a n d eq. (4.18) for the case A ¢ ( I . If we have to~ # 0 the m a p p i n g is also n o t globally reversible. As o n e can c h e c k in fig. 9, w h e r e we m a d e phase portraits o f the m a p (4.20), for co, - 0.1 I, r / - 0.1 and

~3 = "r/+ A, we still have a R i m m e r bifurcation of the type s k e t c h e d in fig. 6, in which an a t t r a c t o r and a repeller are b o r n f r o m the fixed point (x~, y ~ ) - ((L ~1) at a = ( t . In c o n t r a s t to fig. 7 we see in fig. 9 that the K A M curves 10

a r o u n d the F P I do not exist a n y m o r e . This is directly related to the violation of local reversibility at the fixed point. A similar picture can be m a d e for the R i m m e r bifurcation in the m a p p i n g (4.20) for the special case to~ - 0 . In that case the m a p p i n g (4.20) is a reversible m a p p i n g , but the FP1 is an a s y m m e t r i c fixed point. In fact, for ~o~ ¢: 0, the m a p p i n g (4.20) does not have a n o t h e r fixed point for which the e i g e n v a l u e s are the inverses o f the eigenvalues at (x t , y~ ), so then (x~, y~) s h o u l d be a s y m m e t r i c fixed point in o r d e r for the m a p to be reversible. T h e violation of third o r d e r local reversibility shows that the m a p p i n g c a n n o t be reversible.

5. Concluding remarks

In fig. 10, we p r e s e n t two similar pictures of the m a p (4.20) in thc case that ,8 = rb so here the third o r d e r c o n d i t i o n f o r local reversibility is satisfied for A # 0. (It is n o t satisfied at a = 0.) F r o m the figures we m a y infer that also in this case (with w, ¢ 0 ) the m a p p i n g is not reversible. In fact, if the m a p p i n g were reversible the F P I would be a s y m m e t r i c fixed point (and also elliptic for a < 0 ) and in that case o n e expects to o b s e r v e s u r r o u n d i n g K A M curves [15-17] as in fig. 7, r a t h e r than the ( e x t e n d e d ) c h a o t i c structures.

It turns o u t that it takes m u c h m o r e time f o r an orbit to w a n d e r a w a y f r o m the elliptic structure it seems to describe w h e n it is started than in the case of fig. 9, in which the local reversibility is only satisfied up to s e c o n d o r d c r . M o r e generally o n e might anticipate a relation b e t w e e n the time n e e d e d to w a n d c r a w a y f r o m the elliptic structure and the o r d e r of the local reversibility at the FP1. In the f u t u r e we h o p e to give a m o r e detailed analysis of the d y n a m i c a l aspects o f the spiralling of an orbit f r o m a F P I type fixed point.

In fig. 11, finally, we p r e s e n t two p h a s e portraits and the c o r r e s p o n d i n g R i m m e r bifurcation in the case of the m a p p i n g [18]

,r' ( C - y ) ( y ' e - 2 y ' + 2 ) ,

T: y ' x

1 + (y + 1 (,)2 ^I

(5.~)

This m a p p i n g is reversible, i.e. T = I I / 2 with I x a n d 1: given by

T. Post et al. / Bifurcations in 2 D reversible maps 647

0,5952

9,5051

0,5850

- 8 , 8 1

6,5852 b

0,58~1

8,~050

8 8,81

x

-8,81 8 8,81

x ~'

Fig. 10. R i m m e r b i f u r c a t i o n in a n o n - r e v e r s i b l e m a p p i n g . T h e figures are p h a s e p o r t r a i t s of t h e e x a m p l e m a p (4.20) w i t h w 2 = 0 . 1 1 , a n d "q = 6 = 0 . 1 . ( R e c a l l t h a t A = K + 4"rrEy2~.) (a) A t K = - 0 . 5 1 9 5 a n d E = 0.05. H e r e A = - 2 . 2 8 5 × 10 4. W e h a v e a c e n t e r at x = 0. (b) A t K = - 0 . 5 1 9 a n d E = 0.05. H e r e A = - 2 . 2 8 5 × 10 4. W e h a v e a c e n t e r at x = 0. (b) A t K = - 0 . 5 1 9 a n d E = 0.05.

H e r e A ~ + 2 . 7 1 5 × 10 4. W e h a v e a s a d d l e w i t h J = 1 a n d a n a t t r a c t o r a n d a r e p e l l e r . I n b o t h c a s e s w e h a v e local r e v e r s i b i l i t y u p to o r d e r 3. T h e s u r r o u n d i n g o r b i t in b o t h figures s p i r a l s a w a y m u c h s l o w e r f r o m a n e l l i p t i c s t r u c t u r e t h a n in fig. 9, w h e r e w e h a d o n l y s e c o n d o r d e r local r e v e r s i b i l i t y . (a) r e p r e s e n t s o n e s i n g l e o r b i t , w h i l e (b) r e p r e s e n t s t w o orbits.

I, : [x'=y[I+(y'-I), 2 ] ' y , {+ [;-x

^{~))2 12:}

^{{X'y ,}

^{= X ,}

=C-y. _(5.2)

In contrast to the mapping (1.3), the mapping (5.1) and the associated

involutions do not show any singularities. The pictures close to the bifurcation

~4~ T. l'o.s'l el al. B(lk~rca#on.~' in 2 D revcr.viblc map.~

T 1 , 5

fla

• -_

\ }

/

/

, /

/ ~ .. ~ i ¸ /

[ ~ i ¸ ~ [

\ . . j i

I

^{J ./}

^{'i ~\i}

, l

... 1.7;: •

j//

^{/ / /}

t /

1 , 4

I , S ~ . . . J

1 , 6 4 1 , 6 5 1,G6 1 , 6 7

X

I, 5 _ ," - . ~ j

7 jS;5.5.:~A:~.5.::- :: ":: - . . / 1

/ , / / " ,

1 , 4 i ' ", ',\

\ \ . - . ; - .

-%

%-.

1 , 6 4 1 , 6 5 1 , 6 6 1 , 6 7

X

Fig. I I. R i n u n c r bifurcation in ;.i reversible mup. T h e ligurcs tire phase portrails of the e x a m p l e nlap (5.1). (a) At K = 2 . S 2 7 , which is slighlly bclow lhe bifurcation value ( ' : 2 \ ' ~ - 2 . 8 2 8 4 2 7 1 2 5 . . . Wc have a c e n t e r tit (2{(" 2), ( ' / 2 ) . (b) At K 2.83, which is slightly above [he bifurcation wflue (" 2x/2_ 2 . 8 2 8 4 2 7 1 2 5 . . . Wc have a saddle with ~1 1 und an a t t r a c t o r and a r e p e l l e r . In both f i g u r e s (a) and (b) the fixed pohlts are s u r r o u n d e d by K A M curves.

value (7 2C-22 at the symmetric FP1 ( x ~ , 3 , , ) - ( 2 ( ( ' - 2 ) , ( - 7 2 ) arc very similar to fig. 7 with K A M type of curves around the symmetric fixed point.

In general, reversible non-area-preserving mappings can be obtained from reversible area-preserving mappings as follows.

Let T~ v be a reversible area-preserving map, with d e c o m p o s i t i o n T..~e

l a p I l A p 2"

A p p l y an arbitrary coordinate transformation 5 on o n e of the inw)lutions, say on l w , , and obtain

T N A P = S

IIApISIAee,

(5.3)

T. Post et al. / Bifurcations in 2 D reversible maps 649

which will in general not be area preserving. This trick is due to K.M. Pinnow [19]. For example, in (5.1) we have [18]

x

= x ' =

x' Y ' and S: l + ( y - 1 ) 2 ' (5.4)

lAP1:

y ' x [ y , Y"

Finally, the bifurcation scenarios described above can be reproduced in a phase diagram in control-parameter space, that is, one can indicate for instance the regions in the K - e plane in which asymmetric fixed points occur. It would be of much interest to investigate also the regions with higher-period asymmet- ric orbits. This we hope to do in the future.

A n o t h e r point of interest would be to have a closer look at the cross-over properties of Feigenbaum constants and scaling factors close to the symmetric fixed points at which the asymmetric fixed points are born. Such cross-over properties have been studied in refs. [20, 21], in the case of the Hdnon map with (constant) Jacobian ranging from 0 to 1, but here the situation may be slightly more involved.

Acknowledgement

G.R.W.Q. is grateful to the foundation F O M for supporting his visits to the Institute for Theoretical Physics of the University of Amsterdam.

Appendix A

Local reversibility analysis

In this appendix a detailed local reversibility analysis is performed near the FP1 type fixed points of the map (1.3), reading

x ' = x + o J ( y ) ( m o d l ) ,

T: y + K g(x') (A.1)

y,_

1 - eyh(x')

As before, we denote the FP1 by (Xl, y~).

A.1. Expansion around the FP1

For the FP1 type fixed points we have h ( x l ) = g ( x l ) = 0. Furthermore we have ~o(y) E Z. We now expand around such a fixed point. For this purpose we

6511 f . Post et al. Bi¢itrcations in 2 D reversible maps

introduce the translated coordinates

"~--V YI"

(A.2) (A.3)

With the expansions co(y) = bz + o)~z- + co..z g(x) g t w + . w ~ + ~ w ~' h(x) - It~w + h~w: + It~w ~ .

we

obtain for the mapping up to third order in powers of w and z

vt,' w + b z + co~z + co~z , 2

( A . 4 )

(A.5) (A.6)

(A.7)

z ' = w A + z ( l + h A )

+ w ~ [ e y ~ ( - g j ~ g_./l~ + h~) + A(Ey,,h~ + g, 'g~)]

+ 2 w z [ b E y ~ ( - g t ~g~h~ + h : ) + e V,~tl~ + Ab(ey,,h~ + g,

~,~,':)]

+ Z:[Eb:y~( ~i 'g:h, + h:) + 2hey,,h~ -+ A(b'-E)'~,h, + h'-~, t + ,o:)]

+ w : z { 3 b e : y i ' , ( - g , 'g2h~ + h , h : ) + 3 b e y ~ ( - g ~

'g~h,

+

h,)

+2e:y~h~ + 2ey,,h:

+ A[3b~-~,~,h~ + 3t,~y,,( g, '~J,, + I,, ) + 3t, g, '~,, + ~I,, l +wzZ{31,:E-'),,~(--~[ I~,211 ~ 4- ],i/i,) + 3b:~- y~,(--gl Ig, '/1, + h,) +2ey~coAg, 'g~h, + h,) + 4b~'y~,'/, + 4b~.v,,h~ + ~h,

+3b_~ ig~+2bEh~ 2co:(ey,,hl +g, ',~2)]}

+2b:e'-y~,h~ + 2bZey,,h: + b e h , + 2 b e y [ , w : ( - g ~ ~ g 2 h l + h2)

with

T. Post et al. / Bifurcations in 2D reversible maps

+2Ey0o~zh 1

3 2 2 2 1 3 - 1

+A(b E yoh~ + b3Eyog~ g z h l + b3Eyoh2 + b g l g3 + bZEhl +2b~yooozh 1 + 2 b w 2 g ~ l g 2 + w3) ] ,

K 2

A - ~ gl + Eyohl "

The linear part of the map in the new coordinates reads

W t

( Z , ) = (1A 1 b b A ) ( W ) •

651

(A.8)

(A.9)

(A.10)

H = 0. (A.14)

The first condition is also the condition for second order local measure preservation [14]. It evaluates to

o r

The eigenvalue equation is

A A b = ( A - 1) 2 , (A.11)

with A the eigenvalue.

A possible Rimmer bifurcation will take place when we have a single eigenvalue A = 1, that is, when A = 0. We shall now investigate the local reversibility conditions for A = 0 and subsequently for A # 0.

A . 2 . The case A = O

In this case the linear part is already in the Jacobian normal form

1 bl) ' (1.12)

(0

of table I, so we can directly read the coefficients E . . . S from eqs. (A.7), (1.8).

The condition for second order local reversibility is that either

J + 2 E - 2 B H = 0 , (A.13)

652 T. Post et al. / Bifurcations in 2D reversible maps

h I - 0 , (A.15)

which is g e n e r a l l y not satisfied. T h e s e c o n d condition yields

gl l(t12gl - g : h l ) = O. (A. 16)

This is e q u i v a l e n t to

(h(x) ]'

g ' ( x ~ ) \ g ( x ) / , ,~ (I, ( A . 1 7 )

m e a n i n g that the FP1 u n d e r study m u s t be at an e x t r e m u m of h ( x ) / g ( x ) , which is exactly the p r e r e q u i s i t e for the R i m m e r b i f u r c a t i o n to occur.

T h e condition for third o r d e r local reversibility reads ( f o r ~ . - ()) 6 B e P 2 B Q - 6 B L + 2 E K + B E J + 6 B E : + 2 B J e F J - . I K - O .

e v a l u a t i n g to

2 3 I ~ ~

3b ey,,( 2g, g2hlhe + gl-g~h-r + h~) 5 b e y ~ ( g ,

(A.18)

This simply yields

Ig, h~ ^- t l l h 2 ) - 2h~ 0 ( A . 1 9 )

A . 3 . The case a # O

W h e n a # 0 we h a v e two e i g e n v a l u e s A and A ~. and we can t r a n s f o r m the e x p a n d e d m a p p i n g ( A . 7 ) , ( A . 8 ) to the diagonal f o r m

A ()

0 A 1)- (A.21)

F o r this p u r p o s e we a p p l y the diagonalizing t r a n s f o r m a t i o n A Iv + y

w - ( A . 2 2 )

I + A t

A 'b ' ( 1 - A ) ( y - x )

1 + A ~ ( A . 2 3 )

h"(x~) = g " ( x , ) l). (A.2I))

T. Post et al. / Bifurcations in 2D reversible maps 653

T o g e t h e r with the scalings ( 1 - A ) Z A 2b 2

~"Q2 = 1 + A - I 602 '

( 1 - A)3A 3b 2 123 = (1 + A - I ) 2 093 '

E

~ 1 - - ~ I + A h, , 1

d Y ) _ _

"~2=gll I + A 1 g2,

= (1 - A ) - ' e A b

I + A ' g ~ ' ( h z g I - g 2 h , ) ,

1 1

~ = g ~ ( I + A - ~ ) 2 g 3 ,

~ = ( l - A ) leAb

- (1 + d ,)2 g~ '(h3gl - g3hl),

we t h e n find the coefficients E . . . S as E = ~(iAyo

+ ~ [ - 2(2Ay o + ~2A(A - 1) +

I22A21,

F - I+A1 [-2~2y2o + 2 ~ ( A , - 1 ) - 212zA: ] ,

G = - ~ l A ly o

+ ~ 1 [-~(2A-'y~ + ~ 2 ( - A ' + 1) +

S)2A2I,

H -- - 2(ly o

1

+ ~ [~f2Y~ + c~2(1 - A) + g221,

1 -1 2 - 1 ) - 2 0 2 ] ,

J - 1 + ~ [2~2A Yo + 2 ~ 2 ( A - '

(A.24)

( A . 2 5 )

( A . 2 6 )

(A.27)

( A . 2 8 )

(A.29)

(A.30)

( A . 3 1 )

(A.32)

( A . 3 3 )

( A . 3 4 )

(A.35)

654 T. Post et al. .' B(l~rcations in 2 D reversible map.s K = ,~fflA :Yo

1

-Yo +~&A e(l A ) + 1 2 : ] . (A.36)

1

+.'ff~A'-h I ( A - I ) / ( A + 1) 2#e.(2eAv ~+ ;~,Ah ~y,,( A + 1) -R;A)'~+2<~eg2eA(A 1)+ & A ( A 1)-- ~2~A'-b ']. (A.37)

M :

1

=~ I~),L,(=A : - 1 ) + 2 , 7 ( ~ t > v , , ( - ~ A : 1) + ~ , b ~(-A ~ + 2 A e - 2 A + I ) / ( A + 1)+2.:~e~2ey~(2A I) + ;h~e b ~Yo( 1 A) - 3.TQv~ + 2'&.Q:(- 2A e + 3A 1)

+3'.qgA - 1) + 312sA'-h ~] . (A.38)

N = -- :,g ~ A * v:

. 0

1 l 3

+ I ~ A [-37(~.ff~A y,, + 27~ I%~y,~(-2A

"7' 2 2'

+:#'~b ~ ( - A e + 2 A + A ~ 2 ) / ( A + 1)+.,ire.Q:3,,(..---/1 ) +,7(;b ~v,~(1 A ~ ) - 3 # , A ~ v : + 2 < & ~ Q , ( A e - 3 A + 2 )

+ t) + 2;JC,.C22),,,(A: + 2)

+3~&( - A i + l ) M2~A:h i i. (A.39)

+ 1 7 ~

1

[ - )#' .7¢'~A :Yi] 2 'h:A :y,, 2 #,-Q:3',,

+,~l b ~(1 A ~)/(A + l ) - 2,'~:$2~y~ + 7~:A :h ~y,,(A l)

-:,g'~A "-y~ + 25~2~22(A- I ) + ~.O~A "-(A- I )+ ~I~A2h ']. (A.40)

1

+ T ~ A [~'~ ~ 5 ' ~ - 2 ~ ~A)',, - 2g(~ ~Q2A3'o

+ ~'~,Ab ' ( 1 - A ) / ( A + l ) + 2Jt2.(22y~ + . ~ b ' ) , , , ( A - l)

+ 9~3y~ + 2 .~2~2e(1 m ) + . ~ ( l - A ) - ~ 2 , b ']. (A.41)